Problem
Find constant $k\in \mathbb{R}$ in matrix so that matrix $A$ is orthogonal when:
$$ A = \begin{bmatrix} 1 & -1 & -7 \\ 1 & 3 & -1 \\ 2 & -1 & k \end{bmatrix} $$
Attempt to solve
A matrix is orthogonal when $AA^T=A^TA=I$, where $I$ is identity matrix. By utilizing this fact we should be able to solve constant k
$$ \begin{bmatrix} 1 & -1 & -7 \\ 1 & 3 & -1 \\ 2 & -1 & k \end{bmatrix}\begin{bmatrix} 1 & 1 & 2 \\ -1 & 3 & -1 \\ -7 & -1 & k \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
$$ \begin{bmatrix} 51 & 5 & 3-7k \\ 5 & 11 & -k -1 \\ 3-7k & -k-1 & k^2+5 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
It looks like something that could not be solved, but am i simply wrong ?